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| MULTIPLE VORTICES FOR THE SHALLOW WATER EQUATION IN TWO DIMENSIONS |
| Daoming Cao,Zhongyuan Liu |
| (Institute of Applied Math., Chinese Academy of Science, Beijing 100190, PR China;School of Mathematics and Statistics, Henan University, Kaifeng 475004, Henan, PR China) |
| DOI： |
| Abstract: |
| In this paper, we construct stationary classical solutions of the shallow water equation with vanishing Froude number Fr in the so-called lake model. To this end we need to study solutions to a semilinear elliptic problem.We show that if $\frac{q^2}{b}$ has $m$ strictly local minimum (maximum) points $\bar z_i,\,i=1,\cdots,m$, then there is a
stationary classical solution approximating stationary $m$ points vortex solution of shallow water equations with vorticity $\sums_{i=1}^m\frac{2\pi q(\bar z_i)}{b(\bar z_i)}$. Moreover, strictly local minimum points of $\frac{q^2}{b}$ on the boundary can also give vortex solutions for the shallow water equation. |
| Key words: shallow water equation; free boundary; stream function; vortex solution |